Introduction to Cartesian Cubical Sets

This book introduces the category of Cartesian cubical sets and endows it with a Quillen model structure using ideas coming from Homotopy type theory. In particular, recent constructions of cubical systems of univalent type theory are used to determine abstract homotopical semantics of type theory.

Univalence Axiom and Model Structure

The celebrated univalence axiom of Voevodsky plays a key role in establishing the basic laws of a model structure, showing that the homotopical interpretation of constructive type theory is not merely possible, but in a certain, precise sense also necessary for the validity of univalence.

Proofs and Methods

Fully rigorous proofs are given in diagrammatic style, using the language and methods of categorical logic and topos theory.

Intended Audience

The intended readers are researchers and graduate students in homotopy theory, type theory, and category theory.